# Journal of the Ramanujan Mathematical Society

Volume 30, Issue 3, September 2015 pp. 267–294.

Controlled Floyd separation and non relatively hyperbolic groups

**Authors**:
Shubhabrata Das and Mahan Mj

**Author institution:**School of Mathematical Sciences, Department of Mathematics, RKM Vivekananda University, P.O. Belur Math, Howrah 711 202, India

**Summary: **
We introduce the notion of controlled Floyd separation between
geodesic rays starting at the identity in a finitely generated
group G. Two such geodesic rays are said to be Floyd separated
with respect to quasigeodesics if the (Floyd) length of
c-quasigeodesics (for fixed but arbitrary c) joining points on
the geodesic rays is asymptotically bounded away from zero. This
is always satisfied by Morse geodesics. The main purpose of this
paper is to furnish an example of a finitely generated group $G$
such that
1) all finitely presented subgroups of G are hyperbolic,
2) G has an uncountable family of geodesic rays that are Floyd separated with respect to quasigeodesics,
3) G is not hyperbolic relative to any collection of proper subgroups.
4) G is a direct limit of hyperbolic CAT(0) cubulated groups.
5) G has trivial Floyd boundary in the usual sense.
On the way towards constructing G, we construct a malnormal
infinitely generated (and hence non-quasiconvex) subgroup of a
free group, giving negative evidence towards a question of Swarup
and Gitik.

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